Journal of the Ramanujan Mathematical Society

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Some Explicit Computations in Arakelov Geometry of Abelian Varieties


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1 Universite Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France
     

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Given a polarized complex abelian variety (A, L), a Gromov lemma makes a comparison between the sup and L2 norms of a global section of L. We give here an explicit bound which depends on the dimension, degree and injectivity diameter of (A, L). It rests on a more general estimate for the jet of a global section of L. As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and W¨ustholz on one hand and Bost on the other. They also improve some similar statements established by Graftieaux in 2000.
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  • [Au1] P. Autissier, Un lemme matriciel effectif, Math. Z., 273 (2013) 355–361.

  • [Au2] P. Autissier, An upper bound for the theta function, Appendix of Heights on squares of modular curves by P. Parent, Algebra & Number Theory, 12 (2018) 2065–2122.

  • [BG] V. Bosser and E. Gaudron, Logarithmes des points rationnels des vari´et´es ab´eliennes, Canad. J. Math., 71 (2019) 247–298.

  • [Bo1] J.-B. Bost, Intrinsic heights of stable varieties and abelian varieties, Duke Math. J., 82 (1996) 21–70.

  • [Bo2] J.-B. Bost, Periodes et isogenies des varietes abeliennes sur les corps de nombres (d’apr`es D. Masser et G. W¨ustholz), S´eminaire Bourbaki, volume 237 of Ast´erisque, Soci´et´e Math´ematique de France (1996) 115–161.

  • [Ch] K. P. Choi, On the medians of gamma distributions and an equation of Ramanujan, Proc. Amer. Math. Soc., 121 (1994) 245–251.

  • [Ga1] E. Gaudron, Formes lineaires de logarithmes effectives sur les vari´et´es ab´eliennes, Ann. Sci. ´Ecole Norm. Sup., 39 (2006) 699–773.

  • [Ga2] E. Gaudron, Minima and slopes of rigid adelic spaces, Lecture notes of the Summer school Arakelov geometry and Diophantine applications, held at Grenoble (2017). To appear.

  • [GR1] E . Gaudron and G. Remond, Theoreme des periodes et degr´es minimaux d’isog´enies, Commentarii Math. Helv., 89 (2014) 343–403.

  • [GR2] ´E . Gaudron and G. Remond, Polarisations et isog´enies, Duke Math. J., 163 (2014) 2057–2108.

  • [GS] H. Gillet and C. Soul´e, An arithmetic Riemann-Roch theorem, Invent. Math., 110 (1992) 473–543.

  • [Gr] P. Graftieaux, Formal groups and isogeny theorem, Duke Math. J., 106 (2001) 81–121.

  • [Ig] J. Igusa, Theta functions, Die Grundlehren der mathematischen Wissenschaften, Band 194. Springer-Verlag (1972).

  • [Mal] R.-D. Malikiosis, Lattice-point enumerators of ellipsoids. Combinatorica, 33 (2013) 733–744.

  • [Mas] D. Masser, Small values of heights on families of abelian varieties, in Diophantine approximation and transcendence theory (Bonn, 1985). Lecture Notes in Math., 1290 (1987) 109–148.

  • [Mu] D. Mumford, Abelian varieties, Oxford University Press, London (1974).

  • [R´e] G. R´emond, Conjectures uniformes sur les vari´et´es ab´eliennes, Quarterly J. Math., 69 (2018) 459–486.

  • [Via] E. Viada, Slopes and abelian subvariety theorem, J. Number Theory, 112 (2005) 67–115.


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  • Some Explicit Computations in Arakelov Geometry of Abelian Varieties

Abstract Views: 352  |  PDF Views: 0

Authors

Eric Gaudron
Universite Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Abstract


Given a polarized complex abelian variety (A, L), a Gromov lemma makes a comparison between the sup and L2 norms of a global section of L. We give here an explicit bound which depends on the dimension, degree and injectivity diameter of (A, L). It rests on a more general estimate for the jet of a global section of L. As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and W¨ustholz on one hand and Bost on the other. They also improve some similar statements established by Graftieaux in 2000.

References